A341101 T(n, k) = Sum_{j=0..k} binomial(n, k - j)*Stirling1(n - k + j, j)*(-1)^(n-k). Triangle read by rows, T(n, k) for 0 <= k <= n.

1, 0, 2, 0, 1, 4, 0, 2, 6, 8, 0, 6, 19, 24, 16, 0, 24, 80, 110, 80, 32, 0, 120, 418, 615, 500, 240, 64, 0, 720, 2604, 4046, 3570, 1960, 672, 128, 0, 5040, 18828, 30604, 28777, 17360, 6944, 1792, 256, 0, 40320, 154944, 261656, 259056, 167874, 74592, 22848, 4608, 512

Offset: 0

Author

Peter Luschny, Feb 09 2021

Keywords

• nonn
• tabl

Examples

			Triangle starts:
n\k  0     1     2     3     4     5     6     7     8 ...
0:   1
1:   0     2
2:   0     1     4
3:   0     2     6     8
4:   0     6    19    24    16
5:   0    24    80   110    80    32
6:   0   120   418   615   500   240    64
7:   0   720  2604  4046  3570  1960   672   128
8:   0  5040 18828 30604 28777 17360  6944  1792   256
		

Links

• Özmen, N., Erkuş-Duman, E. (2019). On the Generalized Sylvester Polynomials. In: Lindahl, K., Lindström, T., Rodino, L., Toft, J., Wahlberg, P. (eds) Analysis, Probability, Applications, and Computation. Trends in Mathematics. Birkhäuser, Cham. See page 48.

Crossrefs

Alternating row sums: (-1)^n*(n+1) = A181983(n+1).

Cf. A000522 (row sums), A097204 (row sums - 2^n), A002627 (row sums - n!).

Cf. A000166, A340264.

Programs

Maple

T := (n, k) -> add(binomial(n, k - j)*Stirling1(n - k + j, j)*(-1)^(n-k), j=0..k):
seq(print(seq(T(n,k), k = 0..n)), n = 0..9);
# Alternative:
SP := (n, x) -> (x^n)*hypergeom([-n, x], [], -1/x):
row := n -> seq(coeff(simplify(SP(n, x)), x, k), k = 0..n):
for n from 0 to 8 do row(n) od; # Peter Luschny, Nov 23 2022

Mathematica

T[ n_, k_] := If[ n<0, 0, n! * Coefficient[ SeriesCoefficient[ E^(x * z) / (1 - z)^x, {z, 0, n}], x, k]]; (* Michael Somos, Nov 23 2022 *)

PARI

T(n, k) = sum(j=0, k, binomial(n, k-j)*stirling(n-k+j, j, 1)*(-1)^(n-k)); \\ Michel Marcus, Feb 11 2021

PARI

{T(n, k) = if( n<0, 0, n! * polcoeff( polcoeff( exp(x*y) / (1 - x + x * O(x^n))^y, n), k))}; /* Michael Somos, Nov 23 2022 */

Python

from math import factorial
from sympy import Symbol, Poly
x = Symbol("x")
def Coeffs(p) -> list[int]:
    return list(reversed(Poly(p, x).all_coeffs()))
def L(n, m, x):
    if n == 0:
        return 1
    if n == 1:
        return 1 - m - 2*x
    return ((2 * (n  - x) - m - 1) * L(n - 1, m, x) / n
          - (n  - x - m - 1) * L(n - 2, m, x) / n)
def Sylvester(n):
    return (-1)**n * factorial(n) * L(n, n, x)
for n in range(7):
    print(Coeffs(Sylvester(n))) # Peter Luschny, Dec 13 2022

Formula

Sum_{k=0..n-1} T(n, k) = Sum_{k=0..n} binomial(n, k)*(k! - 1) = A097204(n).

E.g.f. for row polynomials: P(x, z) := Sum_{k>=0} T(n, k) * x^n * z^k/k! = e^(x*z) / (1 - z)^x = 1 + (2*x) * z + (x + 4*x^2) * z^2/2! + ... - Michael Somos, Nov 23 2022

From Peter Luschny, Nov 24 2022: (Start)

T(n, k) = [x^k] (x^n)*hypergeom([-n, x], [], -1/x).

T(n, k) = [x^k] (-1)^n * n! * L(n, -x - n, x), where L(n, a, x) is the n-th generalized Laguerre polynomial. (End)